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Question Number 106653 by Dwaipayan Shikari last updated on 06/Aug/20

30+144+420+960+1890+3360+...n

$$\mathrm{30}+\mathrm{144}+\mathrm{420}+\mathrm{960}+\mathrm{1890}+\mathrm{3360}+...{n} \\ $$

Commented by Dwaipayan Shikari last updated on 06/Aug/20

y_0 △y_0 △^2 y_0 △^3 y_0 △^4 y_0 30 114 144 162 276 102 420 264 24 540 126 960 390 24 930 150 1890 540 1470 3360 φ(y_0 )=y_0 +(n−1)△y_0 +(n−1)(n−2)((△^2 y_0 )/(2!))+(n−1)(n−2)(n−3)((△^3 y_0 )/(3!)) +(n−1)(n−2)(n−3)(n−4)((△^4 y_0 )/(4!)) φ(y_0 )=30+(n−1)(114+81n−62+102+8n^2 −85n+n^3 +6n+20n−24) φ(y_0 )=30+(n−1)(n^3 +8n^2 +22n+30)=n^4 +14n^2 +7n^3 +8n Σ_(n=1) ^n φ(y_0 )=Σ_(n=1) ^n n^4 +14n^2 +7n^3 +8n=(1/(20))n(n+1)(n+2)(n+3)(4n+21)

$${y}_{\mathrm{0}} \:\:\:\:\:\:\bigtriangleup{y}_{\mathrm{0}} \:\:\:\:\bigtriangleup^{\mathrm{2}} {y}_{\mathrm{0}} \:\:\:\:\:\:\:\bigtriangleup^{\mathrm{3}} {y}_{\mathrm{0}} \:\:\:\:\bigtriangleup^{\mathrm{4}} {y}_{\mathrm{0}} \:\: \\ $$$$\mathrm{30}\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathrm{114} \\ $$$$\mathrm{144}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{162} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{276}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{102} \\ $$$$\mathrm{420}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{264}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{24} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{540}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{126} \\ $$$$\mathrm{960}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{390}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{24} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{930}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{150} \\ $$$$\mathrm{1890}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{540} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1470} \\ $$$$\mathrm{3360} \\ $$$$\phi\left({y}_{\mathrm{0}} \right)={y}_{\mathrm{0}} +\left({n}−\mathrm{1}\right)\bigtriangleup{y}_{\mathrm{0}} +\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)\frac{\bigtriangleup^{\mathrm{2}} {y}_{\mathrm{0}} }{\mathrm{2}!}+\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)\left({n}−\mathrm{3}\right)\frac{\bigtriangleup^{\mathrm{3}} {y}_{\mathrm{0}} }{\mathrm{3}!} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)\left({n}−\mathrm{3}\right)\left({n}−\mathrm{4}\right)\frac{\bigtriangleup^{\mathrm{4}} {y}_{\mathrm{0}} }{\mathrm{4}!} \\ $$$$\phi\left({y}_{\mathrm{0}} \right)=\mathrm{30}+\left({n}−\mathrm{1}\right)\left(\mathrm{114}+\mathrm{81}{n}−\mathrm{62}+\mathrm{102}+\mathrm{8}{n}^{\mathrm{2}} −\mathrm{85}{n}+{n}^{\mathrm{3}} +\mathrm{6}{n}+\mathrm{20}{n}−\mathrm{24}\right) \\ $$$$\phi\left({y}_{\mathrm{0}} \right)=\mathrm{30}+\left({n}−\mathrm{1}\right)\left({n}^{\mathrm{3}} +\mathrm{8}{n}^{\mathrm{2}} +\mathrm{22}{n}+\mathrm{30}\right)={n}^{\mathrm{4}} +\mathrm{14}{n}^{\mathrm{2}} +\mathrm{7}{n}^{\mathrm{3}} +\mathrm{8}{n} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{{n}} {\sum}}\phi\left({y}_{\mathrm{0}} \right)=\underset{{n}=\mathrm{1}} {\overset{{n}} {\sum}}{n}^{\mathrm{4}} +\mathrm{14}{n}^{\mathrm{2}} +\mathrm{7}{n}^{\mathrm{3}} +\mathrm{8}{n}=\frac{\mathrm{1}}{\mathrm{20}}{n}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)\left({n}+\mathrm{3}\right)\left(\mathrm{4}{n}+\mathrm{21}\right) \\ $$

Commented by Ar Brandon last updated on 06/Aug/20

�� What's this ?

Commented by Dwaipayan Shikari last updated on 06/Aug/20

Newton′s interpolation formula

$${Newton}'{s}\:{interpolation}\:{formula} \\ $$

Commented by Ar Brandon last updated on 06/Aug/20

OK��

Commented by Dwaipayan Shikari last updated on 06/Aug/20

https://en.wikipedia.org/wiki/Newton_polynomial

Commented by Dwaipayan Shikari last updated on 06/Aug/20

https://en.wikipedia.org/wiki/Interpolation

Commented by Ar Brandon last updated on 06/Aug/20

Thanks

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